Theorem bj-dvdemo2 32499

Description: Remove dependency on ax-13 2245 from dvdemo2 4874 (this removal is noteworthy since dvdemo1 4873 and dvdemo2 4874 illustrate the phenomenon of bundling). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dvdemo2	⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-dvdemo2

Step	Hyp	Ref	Expression
1		bj-el 32492	. 2 ⊢ ∃𝑥 𝑧 ∈ 𝑥
2		ax-1 6	. 2 ⊢ (𝑧 ∈ 𝑥 → (𝑥 = 𝑦 → 𝑧 ∈ 𝑥))
3	1, 2	eximii 1761	1 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-pow 4813

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707

This theorem is referenced by: (None)